By tests of the kinds just described among several squares, the mechanic will soon perceive from the several ascertained results that one or the other of the several squares that he is handling is more accurate than all the others, if not absolutely accurate. There still remains the need of a test, however, to prove the absolute accuracy of the particular square which he believes to be about right. On a drafting table, or a smooth board, let him next perform the following experiment, which is one of the several that might be mentioned in this connection: Draw a straight line, AB, say three feet in length, as shown in Fig. 19. This may be done by a straight-edge. Use a hard pencil sharpened to a chisel point. With the compasses, using A and B as centers, and with a radius longer than one-half of AB strike the arcs CD and EF. Then with the straight-edge draw a straight line, GH, through the intersection of the arcs. If the work is accurately done the resulting angles AOH, HOB, BOG, and GOA will be right angles. Lay the square to be tested onto one of these angles, as shown in the illustration, and with a chisel-pointed pencil scribe along the blade and along the tongue. If the lines thus drawn exactly coincide with those first drawn it is satisfactory proof that the square is accurate, and in the same way the square may be placed against one or the other of these right angles in a way to test its interior angle.

The method shown in Fig. 19 anticipates the use of another tool besides the square in making the test. A right angle, however, may be drawn for the purpose described by a method which uses only the square, and which does not require the services of any other tool, or what is the same thing, consider the tool itself to be the figure drawn, and then measure for the purpose of determining the accuracy of the figure.

Various writers have discussed the properties of the right-angled triangle, but we all know that a square erected on a hypothenuse of a right-angled triangle is equal to the sum of the squares erected on the base and perpendicular. This is a well-known mathematical truth, and it may be applied in the tests we are making. Those carpenters who have had occasion to lay out the foundations of houses are well acquainted with the old rule frequently known as “the 6, 8 and 10,” which depends upon the relationship of the squares of the perpendicular and the base to the square of the hypothenuse. Thus the square of 6 is 36, the square of 8 is 64. The sum of 36 and 64 is 100. And the square of 10 is 100. Now let us make application of this rule to test the steel square.

For the sake of accuracy we want to take figures which are as large as possible, so as to reduce the possible error in measurement to the smallest possible dimensions. Let us take for dimensions, 9, 12 and 15 inches. That these will serve is easily demonstrated. The square of 9 is 81. The square of 12 is 144. The sum of these squares is 225, and the square of 15 is 225. Therefore, if the tool that we are testing shows a dimension of exactly 15 inches measured from 9 on the outside of the tongue to 12 on the outside of the blade, as shown in Fig. 20, it will be proof that the square is correct.

It may be somewhat difficult to make a measurement of this kind on the instrument itself, with sufficient accuracy to be beyond dispute. I suggest, therefore, that the square be laid flat upon an even surface, like a drawing table, and that with a chisel-pointed pencil lines be scribed along the tongue and along the blade. Mark accurately the distance of 9 inches from the heel up the tongue, and 12 inches from the heel along the blade. Then measure diagonally and see if the distance is exactly 15 inches.

In what has preceded there has been a suggestion that the error due to lack of precision in measurement is diminished if the figures are increased in size. If the size of the drafting table permits, therefore, extend the line drawn along the tongue of the square to 3 feet. Extend that drawn along the blade to 4 feet. In doing this care must be taken that the lines thus extended are fair to the tool under examination, for if they are not drawn in a way to strictly coincide with the edges of the square then the test is of no avail. Then measure from the ends of these lines, that is, from a point 3 feet from the heel up the tongue to a point 4 feet from the heel along the blade. If this diagonal distance is exactly 5 feet it will show that the angle represented by the heel of the square, as I have described it, is a right angle, and that, therefore, the test is accurate.

Now let us next examine a little more carefully the relationship of the square to frequently required lines. It is a common thing among carpenters to use 12 of the blade and 12 of the tongue for a right angle or square miter. Why are these figures employed, or to put the question otherwise, how is it determined that 12 and 12 are the proper figures? Perhaps the question can be made still clearer by another illustration. It is common to say that 12 of the blade and 5 of the tongue is correct for the octagon miter. How is this determined? In Fig. 21 there is shown a quarter circle, XG, described from the center C. Along the horizontal line, AB, the blade of the square is laid with 12 of the blade against the center C, from which the quadrant was struck. Now if we divide this quadrant into halves, thus establishing the point E, and if from E we draw a line to the center C, which is 12 of the blade, it will be found that it cuts also 12 of the tongue. If we complete the figure by erecting a perpendicular line from the point X, and intersecting it with a horizontal line from G, thus establishing the point O, it becomes very evident that CE is the miter line of a square.

If we bisect XE, thus establishing the point D, and by the conditions existing setting off in the quadrant a space equal to one-quarter of its extent, and if from D we draw a line to the center, C, corresponding, as already mentioned, with 12 on the blade, we shall find that this line (DC) cuts the tongue on the point 5 (very nearly, the exact figures being 4 31-32 inches). The line DC, as above explained, bisects the eighth of a circle. In other words, it is the line of an octagon miter, and therefore, we say that for an octagon miter we take 12 on the blade and 5 on the tongue.

By dividing the quadrant into three equal parts, as shown by XG, GH and HG, we obtain by drawing GC the line corresponding to the hexagon miter. This, it will be observed, cuts the tongue of the square at 7 (very nearly, the exact figures being 6 15-16 inches), and, therefore, we say for hexagon miters we take 12 of the blade and 7 of the tongue.

The question sometimes arises, can the square be employed to describe a circle? While the square may be used for describing a circle of any diameter, providing the capacity of the square is not exceeded, still those who attempt to perform the work will very likely conclude before they are through that other means are more satisfactory for regular use. The way to proceed is indicated in Fig. 22. Let it be required to describe a circle, the diameter of which is equal to ED. Drive pins or nails at these points and place the square as shown in the sketch. Place a pencil in the interior angle of the square, as shown at F. Then gradually shift the square so that the pencil will move in the direction of D, always being careful to keep the inside of the blade and inside of the tongue in contact with the pins or nails, ED. After having described the arc from F to D reverse the direction describing the arc from F to E. Then turn the square over and by similar means complete the other half of the circle.

THE STEEL SQUARE AND ITS USES.
Division B.
Introductory.

Having dealt with the more simple matters that can be dealt with by aid of the Steel Square, we now take up some of the more difficult problems that can be solved by aid of this useful tool.

Among the problems and solutions offered, are those of laying out braces, having regular or irregular runs, rafters, and roofing generally, ascertaining the length of hips, their bevels, cuts, pitches and angles, jacks, cripples, ridges, purlins, collar beams, and much other matter pertaining to hip or cottage roofs.

Gables, or saddle roofs are dealt with, also mansard roofs, taper framing, odd bevels, splays and other similar work.

I introduce in this division a few remarks regarding the fence made use of when laying out rafters, stairs or other bevelled work. The department also shows how to lay-out stair strings by aid of the square, and many other things that will be found useful to the general workman.

A very good fence for the square may readily be made from a stick of hardwood (Fig. 23) about two inches wide, one and a half inches thick and two and a half feet long. A saw kerf, into which the square will slide, is cut from both ends leaving about 8 inches of solid wood near the middle. The tool is clamped to the square by means of screws at convenient points as shown. Another style of fence, which is made of a piece of hardwood, has a single slot only as shown in Fig. 24. The square is slipped in and fastened in place by screws similar to the first. An application of the fence and square combined is shown at Fig. 25, where the combination is used as a pitch-board for laying out stair strings. In this example the blade is set off at 10 inches, which makes the tread, and the tongue shows the riser, which is set off at 7 inches. The dotted line, ce, shows the edge of the plank from which the string is cut, and h shows the fence, a shows the bottom tread and riser. In this example the riser shows the same height as the riser above it, namely, 7 inches. This is wrong, as the first riser should always be cut the thickness of the tread less than those above it, as shown by the dotted lines on the bottom of the string, then when the tread is in place it will be the same height from the top of the floor to the top of the first tread, that the top of first tread is to top of second one and so on.

Suppose we wish to lay out a rafter having eight inches rise and twelve inches run. Set the fence at the 8″ mark on the blade, Fig. 26, and at the 12″ mark on the tongue, clamping it to the square with 1¼″ screws. Applying the square and fence at the upper end of the rafter we get the plumb-cut P at once. By applying the square as shown twelve times successively the required length of the rafter and foot-cut B is obtained. In this case the twelve applications of the square are made between the points P and B. Run and rise must also be measured between these points. If run is measured from the point B, which will be the outer edge of the wall plate, it will be necessary to run a gauge line through B parallel to the edge of the rafter, and subtract a distance from the height of the ridge to give us the correct rise. The square must then be applied to the line L. A rafter of any desired rise and run may be laid off in this manner by selecting proportional parts of the rise and run for the blade and tongue of the square. For a half-pitch roof use 12 in. on both tongue and blade, for a quarter-pitch use 6 in. and 12 in., for a third-pitch use 8 in. and 12 in., etc. The terms half-pitch, quarter-pitch, etc., refer to the height of the ridge expressed as a fraction of the span.

The line L is supposed to represent the path of the fence as it is slid along the edge of the rafter. This will be explained at greater length in the following pages.

At Fig. 27 I show a method of laying out a rafter without making use of a fence. In this case the roof is supposed to be half-pitch, so we take 12 and 12 on the square and apply it to the rafter as many times as there are feet in half the width of the building, which in this case will be 15 feet, as we suppose the building to be 30 feet wide. As the lower end of the rafter is notched to sit on the plate we must gauge off a backing line, as shown, to run into the angle of the notch. This line will be the line on which the gauge points 12 and 12 on the square must start from each time.

Starting from this notch apply the square, keeping the twelve-inch mark on both sides of the square carefully on the backing line, and marking off the rafter on the outside edges of the square. Repeat this until you have fifteen spaces marked off, then set back from your last mark half the thickness of the ridge-board, and with the square as before mark off the rafter. This will be the exact length and also the plumb-cut to fit the ridge-board. Or if we take the diagonal of 12 by 12, which is 17, and mark off 15 spaces of 17 in., making the necessary allowance for the half thickness of the ridge-board, it will amount to the same thing, every 17 in. on the rafter being nearly equal to one foot on the level.

Should the building measure 30 ft., 9 in. in width—the half of which is 15 ft., 4½ in.—we take the fifteen spaces of 12 by 12 and then the 4½ in. on both sides of the square on the backing line as before. This will give us the extra length required. The same rule will apply to any portion of a foot there may be.

A fence, sometimes called a stair gauge, is manufactured of metal by the Cheney & Tower Company, Athol, Mass., which I show at Fig. 28, and is considered about the best thing of the kind. It consists of a piece of polished angle metal, each side being ⅞ inch wide. One side is slotted to accommodate the heads of the set-screws and to allow the slides to be fastened at the desired points. The gauge is fastened to any square and is useful for laying out stairs, cutting in rafters, cutting bevels or other angles. In marking off stairs with an 8-inch rise and an 11¾-inch tread the gauge would be fastened at 8 inches on one end of the square and 11¾ at the other end. The square would then be laid on the plank with the face of the gauge against its edge and the mark made around the point of the square. This would be repeated until the required number of steps were marked. The gauges are made in two sizes, 18 and 28 inches long. It is stated that mechanics who have used it find it one of the handiest tools in their kits.

Another style of fence is shown at Fig. 29 in conjunction with a slotted square. This, perhaps, is the handiest of all the devices for a fence, but it is expensive, and as constructed requires a square with a slot in each arm, and as a rule workmen do not take kindly to squares with slots in them. A shows the square, B the fence, SS set screws to hold the fence in position, and ff the points of the square.

The application of the square and fence combined for laying out a housed string for stairs is shown at Fig. 30. In this example the fence is a single slotted one, and three screws are employed to hold the square in position. The rise is seven inches and the tread is laid off nine inches on the blade. The square at the foot of the string shows how the latter should be finished to make the floor and the base-board. In case no pitch-board is required, as the square when adjusted with fence, as shown, does the work of the pitch-board.

There are many other applications of the fence in connection with the square that I may have cause to refer to as I proceed, as it is my desire to present in this work everything I can collect regarding the square that I think will be of service to the workman. Doubtless there will be many descriptions and illustrations some of my readers will have met with before, or which they have been acquainted with for a long time. The great bulk of readers, however, will be new hands and unacquainted with the use of the square beyond its simple application as a squaring tool, and what may appear to be a useless rule to the expert or old hand will prove a choice tidbit to the beginner and will whet his appetite for further knowledge on the subject. Indeed this book is prepared more particularly for the younger members of the craft, although a majority of the older workers will find much in it that will interest, amuse and instruct.

It will be seen that the fence or guide used in connection with the square is, after all, a very simple matter, and would, no doubt, suggest itself to any clever workman who was laying off rafters with the square.

BRACE RULES.

It will now be in order to show how the square can be used for getting the lengths and bevels for braces of regular and irregular runs. If we wish to lay out a brace having a three-foot run on both post and beam, the matter is quite simple, for we can take 12 inches on the tongue and 12 inches on the blade and transfer this distance three times on a straight line and we have the extreme length of the brace from point to point, and by marking along the blade at one end of this length and along the tongue at the other end we also get the bevels. This is easy and simple enough, and without further refinement will give the lengths and bevels exactly for a flat-footed brace. When the run is different than the rise, as in the example shown at Fig. 31, the square is applied in a somewhat different manner. Here we have a run of three feet and a rise of four feet. To get the proper length and bevels for a brace to fit in this situation we must use 12 inches on the tongue and 16 inches on the blade, then the bevel of the upper end of the brace will be found along the line of the tongue, and the line of the blade will give the bevel for the foot of the brace. In this case the square is transferred three times, just as though the rise and run were both three feet; the difference being made by dividing the odd foot into three equal parts of 4 inches each and adding one part to the blade, thus making the gauge point on the blade 16 inches instead of 12 inches, which regulates the extra length and the change in bevels. A little study on the part of the reader will reveal to him how the square may be set to gauge points so as to make a brace suitable for any rise and run of any right-angled frame work.

A brace intended for equal run and rise of four feet is shown at Fig. 32. Here we have the fence in use, and the square is shown in all its positions from start to finish in the formation of the brace. The gauge line marked 0000 is the line from which the marks 12 and 12 are supposed to measure, and this when squared over as shown leaves a butt, or “heel of the brace,” which is to rest on a shoulder “boxed” in both beam and post. The dotted lines on the ends of the brace show the tenons for which mortises are made in both post and girt or beam. It must be understood, of course, that this operation is only performed once for each kind of brace, and that on a pattern made of some kindly wood, such as pine, cedar or whitewood. For the pattern, dress up a piece of wood to 4 inches wide if the braces are to be made of 4x4-inch stuff; if for larger or smaller stuff then make the pattern the width of brace to suit. Have the pattern of sufficient length; if for a 4-foot run and rise it will require to be not less than 6 feet long. Run a gauge line three-eighths of an inch from the straight or front edge, as shown at 0000, and set the two 12-inch marks on this line, then screw the fence tight on the square with its sliding edge against the edge of the pattern, and then slide and mark as shown four times, when the length and bevels of the brace will be obtained. Provide for the tenons beyond the lines shown by the square, or for a “flat-foot” brace, saw the timber off on the lines shown on the edge of the square. After the pattern is made the fence and square may be laid aside, as the pattern can be used for any number of braces, and when finished with on one job, may be safely placed away to use again for the same “run and rise” when occasion arises. The pattern may be any thickness from half an inch to one inch. The same rules may be observed in making patterns for any regular or irregular runs and rises.

With regard to the brace rule as given on steel squares, I may say that there is some slight difference in the lengths given by different makers—though nearly all modern makes figure up alike—but this difference is so small that in soft wood framing it has no effect. In hardwood framing the framer never applies these rules, but gets his lengths with the square and fence.

The length of any brace simply represents the hypothenuse of a right-angled triangle. To find the hypothenuse, extract the square root of the sum of the square of the perpendicular and horizontal runs. For instance, if 6 feet is the horizontal run and 8 feet the perpendicular, 6 squared equals 36, 8 squared equals 64, 36 plus 64 equals 100, the square root of which is 10. These are the figures generally used for squaring the frame of a building or foundation wall.

If the run is 42 inches, 42 squared is 1764, double that amount, both sides being equal, gives 3528, the square root of which is, in feet and inches, 4 feet, 11.40 inches.

In cutting braces always allow in length from a sixteenth to an eighth of an inch more than the exact measurement calls for.

Directly under the half-inch marks on the outer edge of the back of the tongue will be noticed two figures, one above the other. These represent the run of the brace, or the length of two sides of a right-angled triangle; the figures immediately to the right represent the length of the brace or the hypothenuse. For instance, the figures 36-36 59-91 show that the run on the post and beam is 36 inches, and the length of the brace is 50.91 inches.

Upon some squares will be found brace measurements given where the run is not equal, as 18-24 30. It will be noticed that the last set of figures are each just three times those mentioned in the set that are usually used in squaring a building. So if the student or mechanic will fix in his mind the measurements of a few runs, with the length of braces, he can readily work almost any length required.

Take a run, for instance, of 9 inches on the beam and 12 inches on the post. The length of brace is 15 inches. A run, therefore, of 2, 3, 20, or any other number of times the above figures, the length of the brace will bear the same proportion to the run as the multiple used. Thus, if you multiply all the figures by 4 you will have 36 and 48 inches for the run, and 60 inches for the brace, or to remember still more easily, 3, 4 and 5 feet, or 6, 8 and 10 feet.

There are other runs that are just as easily fixed in the mind. 51-inch run, brace 6 feet, 12 hundredths of an inch; 8 feet, 3-inch run, brace 11 feet, 8 inches, etc.

The following examples and explanations on roof framing are simple and easily understood, and cannot fail of being valuable to the young mechanic who aspires to become an expert roof framer. These examples will serve as starters, and in the following volume, which will be issued shortly, more advanced examples will be presented.

ROOF FRAMING.

Roof framing can be done about as many different ways as there are mechanics. But undoubtedly the easiest, most rapid and most practical is framing with the “square.” The following cuts will illustrate several applications of the square as applied to roof framing, and all who are interested in the subject can, by giving it a careful study, be able to frame any ordinary roof the mechanic comes in contact with.

Fig. 33 is an illustration that could well be given much thought and study. It not only gives the most common pitches, but also gives the degrees.

Most carpenters know that half-pitch is 45 degrees, yet few know third pitch is nearly 34, and quarter-pitch about 27 degrees.

A building 24 feet wide (as the rafters come to the center) has a 12-foot run and half-pitch the rise would also be 12 feet, and the length of the rafter would be 17 feet (the diagonal of 12). Length, cuts, etc., could all be figured from the one illustration.

Fig. 34 illustrates a way to cut rafters with the square.

A roof 14 feet wide would have a run of 7 feet, third-pitch would rise 8 inches to every foot run. Therefore, place the square on 8 and 12 seven times, and you have length and cuts.

Fig. 35. For the octagon rafter, proceed same as common rafter, only use 13 for run (in place of 12 for common rafter).

Fig. 36, hip or valley rafter. As these rafters run diagonal with the common rafter and as the diagonal of 1 foot is practically 17 inches, use 17 for run, and proceed same as common rafter.

Length of jacks. If there are to be five, divide the common rafter into six equal parts, use that for a pattern, and it gives the length very nicely. But that will not always work. To get all the different lengths might at first look difficult even to many good mechanics, but it is very simple as illustrated in Fig. 37. If the first jack was one foot from corner apply the square same as for common rafter, and it gives length and cut (mark the length for starting point on next), and if it is 17 inches from the other move the square up to 17, if the next is 15 move up to 15 and so on.

Fig. 38. The side cut of jack to fit hip, if laid down level would, of course, be square miter, but the more the hip rises the sharper the angle. Measure across the square from 8 to 12, and it is nearly 14½, which is the length of rafter to one foot of run. Length and run, cut on length, gives the cut.

Fig. 39, octagon jack. As the octagon miter on level surface is 5 and 12, it must raise same as common jack, and is, therefore, raised to length, or 14½, and 5 cut on length.

Fig. 40, hip rafter, is also length and run, cut on length.

Fig. 41. To bevel top of hip take length and rise and mark on rise.

Fig. 42 is another practical way, which is simply to lay the square on heel or hip. The illustration, explains itself.

Perhaps the most practical way of all to frame a roof, the simplest to understand, easiest to remember, and most rapid to apply is simply to always take the rise and run, measure across the square which gives length. Rise and run give cuts, so you have it all.

Fig. 43 illustrates a roof 25 feet wide and a rise 10 feet, 9 inches, run 12 feet, 6 inches. Measuring across the square from 10¾ to 12½ gives 16½, or 16 feet, 6 inches is the length of rafter.

Fig. 44. If the run of common rafter is 12½, the run of the hip will be diagonal of 12½ which is 17 8-16, as is plainly illustrated.

Fig. 45. As the rise is 10¾ and run 17 8-12, the length will be 20 feet, 2 inches.

Fig. 46. When a roof must go to a certain height to strike another building at a given point, as in additions, porches, etc., don’t forget in getting the rise from plate to given point to allow the squaring up of heel as illustrated; and also remember to allow for ridge whenever one is used.

Fig. 47 illustrates the cut of top of quarter-pitch rafter to lay on top of roof just mentioned. To apply the square first lay it on 12 and 6, which is quarter-pitch, and gives plumb-cut. From plumb-cut lay off pitch of main roof 10¾ and 12½, which gives cut.

Anyone that has studied this with determination will have no trouble in framing any ordinary roof, as the general principles apply to all roofs, pitches, etc. So I will not take up any more space with roof framing at this time, but remember all sheathing, studding, cornice, etc., are made on the same cuts. In fact a hopper is also exactly on the same principle.

Division B.
SOME POINTERS ON ROOF FRAMING.

No matter what people may say to the contrary, there is no method or methods that has ever been devised that is so effective in roof framing, or results so rapidly achieved, as those which are obtained by the use of the steel square. I have shown in some of the earlier pages of this work how rapidly the length, and bevels of any common rafter may be obtained by the simple application of the square, any determined number of times. Thus for a building of, say, 30 ft. in width, which is to have a roof of any given pitch, we arrange the pitch as I have shown, with so many inches on the blade for the run, and so many on the tongue for the rise. This settled, we apply the square fifteen times to the rafter, 15 being half of the width of the building. This then gives the length of the rafter, and a line drawn along the edge of the tongue of the square will give the proper bevel for the top or plumb cut. If there is to be a ridge board on the roof, then half the thickness of such board must be measured back on the line drawn, and the rafter must be cut at that point, this provides for the ridge board being nailed on the face of the cut without in the least changing the pitch.

A line along the edge of the blade, gives the proper bevel for the level or horizontal cut. If the bottom end of the rafter is to have a crow-foot cut on it to fit the plate, the workman will have no difficulty whatever in cutting the foot of the rafter to suit, as all the lines will be at right angles to each other, and a section of the plate may be made on the line of the bevel and the “cuts” laid off to suit the conditions.

In reviewing an article of mine on this method of laying out a rafter, an English carpenter took exceptions to it on the grounds that it would take too much time to lay out the rafters for a whole building by this “tiresome process,” as he called it. Now the Englishman was right from his point of view, but no American workman would ever think of laying out the rafters for a whole building by the process. He would simply make one rafter as I have shown, for a pattern, and use this pattern for laying out all the other rafters for that particular pitch and rise on the same roof. Most workmen, however, make a pattern from thin stuff of some sort, as it is lighter and easier handled. The reviewer suggested as a better way “that the pitch be arranged on the iron square, then measure across the angle from the points of run and pitch, and multiply this measurement by half the width of the roof to be covered.” Now this is all right, but, as a matter of fact, entails more labor of a “tiresome sort” and would use much more time than the method I have taught now for nearly forty years. The American workman, however, does not even require a suggestion as to the quicker method. He will see and adopt it at once without argument.

The method the Englishman would adopt is shown at Fig. 48, where the points of pitch and run are shown at 12 and 8, which makes the diagonal line 14½ inches. To get the length of the rafter for our supposed building then, we must multiply this 14½ inches fifteen times, then we must use the square at the top and bottom of the timber to obtain the necessary bevels for the cutting lines.

Regarding this question of preparing rafters for a common roof, an “old hand” in the use of the steel square writes to me to say: “I do not think that any simpler method can be given for finding the bevels at the heel and point of rafters than that which you have explained in your books, but I do think that the following method for obtaining lengths of rafters, is somewhat better than yours, particularly when employed for estimating purposes. The most common width of buildings in my locality is 24 ft., and with your permission I purpose to take that width for the practical test of my method. As you have given several ways by which the same result can be obtained, I will ask you to compare them with mine.

Finding the length of the hypothenuse by the old rule, we obtain for one-quarter-inch pitch 13:4.99, or, as near as it can be used on the square 13 feet, 5 inches.

Allowing one inch to the foot and trying your method we find, as a result, 13 inches and 7-16 scant, or 13 feet, 5 inches. This is a very simple method, and when the rule is kept perfectly straight, the results are very satisfactory.

By my way I simply multiply the width of the building by the decimal .56, 24×.56=13.44, or as near as can be worked by the square, 13 feet, 5 inches.

Let us try the same rule for a greater width—say 60 feet. By finding the hypothenuse we find as near as can be used by the square, 33 feet, 6½ inches. By my method it would be 60×.56, or 33.60, equal to 33 feet, 7 inches full. By this method the rafters in wide buildings are a little long. Thus, if the building is 52 feet wide, by the hypothenuse it would be 29 feet, 1 inch; my way it would be 29 feet, 1½ inches. I consider this an advantage, as it leaves the point of the rafter very slightly open.

For one-third I follow the same plan, only using the decimal .6. Unlike the decimal used for a quarter pitch the lengths are a very small fraction short; as, for instance, a rafter for a building 60 feet wide, by finding the hypothenuse, would be 36 feet, 1-16 of an inch. By my way, 60×.6=36 feet. A slight difference, truly. If building is 48 feet wide, then by the first method we find 28 feet, 10 inches full; by my way, 28 feet, 9⅜ inches. A little practice will enable the mechanic to allow just enough to make up for the slight difference, so that when rafters are put together the fit will be perfect.

The one-half pitch can be found in the same manner by using the decimal .71. Taking the 24-foot building, length of rafters by the hypothenuse, we find 16 feet, 11 2-3 inches; my way they would be 17 feet full. Again, building 60 feet wide, rafters by the first method would be 42 feet, 6⅛ inches; by my way 60×.71=42 feet, 6 inches. By using this decimal, the length is so near practically correct, that it may be used in all cases.

For a full pitch use the decimal 1.12, and as in the preceding mentioned pitch, and it will be found so near correct that it can be practically used in all cases.

It will be noticed that I have not made any allowance for projection of rafters over the plate. In this case gauge from the crowning side of your rafter the thickness of your projection; allow enough for the latter, and find the lower bevel according to the way you described in your last; measure the length of your rafter from where this bevel crosses the gauge line.

A little practice will enable the mechanic to lay off a rafter in a very short time. I have used the above myself, and have no trouble whatever. While I have no fault to find in your methods, as I know them to be correct, yet it is just as well that workmen should know other methods, as there are many occasions when the “only method” he possesses cannot be applied. Hence I submit the foregoing, at your request.

W. H.”

All this is very true, and right as far as it goes, but it so happens that many workmen do not have the necessary learning to work out these problems in footing on the lines laid down by W. H., but, in order to meet conditions of this kind I have prepared a series of tables which is inserted in the larger volumes, giving the length of rafters for any building having a width of from five to sixty feet and a rise of roof of from one to eighteen feet to ridge. This will cover the whole ground, and form a ready table for the estimator to take his quantities from.

I may be pardoned for again showing the common and simplest method of laying out an ordinary rafter, for notwithstanding all I have said and described and explained on this subject, there will always be some persons who will not be able to grasp the method, unless it is put to them in some other light. I am sure of this from the long experience I have had in the answering of questions of this kind through the columns of different building journals. This is no doubt owing to some constitutional peculiarities of both the person who makes the inquiry and the person who attempts to answer it. This is one of the main reasons why I have admitted into this work various methods and descriptions of others than myself, so that readers will have the same methods described and explained to them in several different ways by several writers.

Let us take the diagrams shown at Fig. 49, which shows a portion of a roof having a quarter pitch. CEB showing the height, and AB the length and inclination of rafter. D shows the foot of the rafter on the plate, cut “flat foot” and the line EC the plumb cut. This is quite plain. The building may be any width, let us say in this case, that it is 30 feet wide from A to O. That will make the distance from A to C 15 feet.